Great suggestions by @andrjohns and @jsocolar. Just in case it’s relevant, if you want to constrain the product of more than two positive parameters \beta to be equal to some positive value \theta, you can use a simplex parameter \kappa of length equal to the number of parameters values in \beta, and define \beta_i = \theta^{\kappa_i}. Because \Pi_i \beta_i = \Pi_i \theta^{\kappa_i} = \theta^{\Sigma_i \kappa_i} and \Sigma_i \kappa_i = 1 (because it’s a simplex), your product of parameter values in \beta are guaranteed to be \theta.
Note that this will only work for target values of the product other than 1.0.